Season & Sear the Chicken: Pat chicken cutlets dry with paper towels. Season both sides evenly with kosher salt, freshly cracked black pepper, and a light dusting of flour. Heat 1 tablespoon olive oil in a 12-inch oven-safe cast iron skillet over medium-high heat. Sear cutlets for 3 to 4 minutes per side until golden brown. Transfer chicken to a clean plate (cutlets will finish cooking in the oven).
Caramelize the Onions: Lower skillet heat to medium. Add butter and remaining olive oil to the pan drippings. Add sliced onions and 1/4 teaspoon salt. Cook for 20 to 25 minutes, stirring occasionally, until onions melt down into a sweet, deep golden jammy brown. (Splash 1 tablespoon of broth if pan becomes dry).
Build the French Onion Pan Gravy: Stir in minced garlic and cook for 1 minute. Pour in balsamic vinegar to deglaze the pan, scraping up all golden caramelized bits from the bottom. Whisk in beef bone broth, Dijon mustard, and fresh thyme leaves. Bring to a gentle simmer for 3 minutes until slightly thickened.
Nestle Chicken & Smother: Preheat oven broiler to HIGH (or 425°F/220°C bake). Nestle seared chicken cutlets back into the skillet directly into the bed of caramelized onions. Spoon generous ladlefuls of onions and pan gravy over the top of each cutlet.
Blanket with Cheese & Broil: Scatter the shredded Gruyère and Swiss cheese generously over each smothered cutlet. Transfer skillet to the oven and broil for 3 to 4 minutes until cheese is completely melted, bubbling, and spotted with golden brown crust.
Garnish & Serve: Remove skillet carefully with an oven mitt. Garnish with fresh thyme leaves and cracked black pepper. Spoon the glistening onion pan juices around the cutlets and serve immediately with garlic mashed potatoes or crusty baguette.